A connection between extremely strong damped Lyman-

We present a study of ~100 high redshift (z ~ 2−4) extremely strong damped Lyman-α systems (ESDLA, with N(H i) ≥ 0.5 × 1022cm-2) detected in quasar spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (SDSS-III) Data Release 11. We study the neutral hydrogen, metal, and dust content of this elusive population of absorbers and confirm our previous finding that the high column density end of the N(H i) frequency distribution has a relatively shallow slope with power-law index −3.6, similar to what is seen from 21-cm maps in nearby galaxies. The stacked absorption spectrum indicates a typical metallicity ~1/20th solar, similar to the mean metallicity of the overall DLA population. The relatively small velocity extent of the low-ionisation lines suggests that ESDLAs do not arise from large-scale flows of neutral gas. The high column densities involved are in turn more similar to what is seen in DLAs associated with gamma-ray burst afterglows (GRB-DLAs), which are known to occur close to star-forming regions. This indicates that ESDLAs arise from a line of sight passing at very small impact parameters from the host galaxy, as observed in nearby galaxies. This is also supported by simple theoretical considerations and recent high-z hydrodynamical simulations. We strongly substantiate this picture by the first statistical detection of Ly α emission with ⟨LESDLA(Ly α)⟩ ≃ (0.6 ± 0.2) × 1042 erg s-1 in the core of ESDLAs (corresponding to about 0.1 L⋆ at z ~ 2−3), obtained through stacking the fibre spectra (of radius 1 ″ corresponding to ~8 kpc at z ~ 2.5). Statistical errors on the Ly α luminosity are of the order of 0.1 × 1042 erg s-1 but we caution that the measured Ly α luminosity may be overestimated by ~35% due to sky light residuals and/or FUV emission from the quasar host and that we have neglected flux-calibration uncertainties. We estimate a more conservative uncertainty of 0.2 × 1042 erg s-1. The properties of the Ly α line (luminosity distribution, velocity width and velocity offset compared to systemic redshift) are very similar to that of the population of Lyman-α emitting galaxies (LAEs) with LLAE(Ly α) ≥ 1041erg s-1 detected in long-slit spectroscopy or narrow-band imaging surveys. By matching the incidence of ESDLAs with that of the LAEs population, we estimate the high column density gas radius to be about rgas = 2.5 kpc, i.e., significantly smaller than the radius corresponding to the BOSS fibre aperture, making fibre losses likely negligible. Finally, the average measured Ly α luminosity indicates a star-formation rate consistent with the Schmidt-Kennicutt law, SFR (M⊙ yr-1) ≈ 0.6 /fesc, where fesc < 1 is the Ly α escape fraction. Assuming the typical escape fraction of LAEs, fesc ~ 0.3, the Schmidt-Kennicutt law implies a galaxy radius of about rgal ≈ 2.5 kpc. Finally, we note that possible overestimation of the Ly α emission would result in both smaller rgas and rgal. Our results support a close association between LAEs and strong DLA host galaxies